If every vertex of a triangle is joined by straight lines to 6 points on
the opposite side of the triangle, how many regions are formed? If every
vertex of a regular pentagon is connected to every other vertex, how many
triangles are formed?

In an ellipse with major axis of 2a, minor axis of 2b, and foci c (on the
major axis), the relationship c squared = a squared - b squared holds
true... how do the three numbers fit into a Pythagorean relationship?

Within triangle ABC, draw three segments parallel to the sides of the
triangle, each touching two sides. The three segments meet at one
point, and they are all the same length, x. Find the length of x given
the length of the sides of triangle ABC.

A student wonders if there is an equation for a rectangle in the Cartesian plane,
comparable to the one for the equation of a circle. Doctor Peterson confirms the student's
suspicions about the difficulties involved.